The derivative of (log x)^{x} with respect to | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let $u = (\log x)^{x}$.Taking logarithm on both sides,$\log u = x \log(\log x)$.Differentiating with respect to $x$,we get $\frac{1}{u} \frac{du}{

Vedclass · Accepted Answer

$x(\log x)^{x}\left[\frac{1}{\log x}+\log (\log x)\right]$

Vedclass · Answer

(C) Let $u = (\log x)^{x}$.Taking logarithm on both sides,$\log u = x \log(\log x)$.Differentiating with respect to $x$,we get $\frac{1}{u} \frac{du}{dx} = x \cdot \frac{1}{\log x} \cdot \frac{1}{x} + \log(\log x) \cdot 1$.So,$\frac{du}{dx} = (\log x)^{x} \left[ \frac{1}{\log x} + \log(\log x) \right]$.Let $v = \log x$. Then $\frac{dv}{dx} = \frac{1}{x}$.We need to find $\frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{(\log x)^{x} [ \frac{1}{\log x} + \log(\log x) ]}{1/x}$.Therefore,$\frac{du}{dv} = x(\log x)^{x} \left[ \frac{1}{\log x} + \log(\log x) \right]$.

The derivative of $(\log x)^{x}$ with respect to $\log x$ is

Similar Questions

If $y = \tan x \tan 2x \tan 3x$,then $\frac{dy}{dx}$ is equal to:

If $y=(1+x)(1+x^{2})(1+x^{4})$,then $\frac{dy}{dx}$ at $x=1$ is

If $x^{m} y^{n}=(x+y)^{m+n}$,then $\frac{dy}{dx}$ is equal to

Differentiate $\sqrt{\frac{(x-3)(x^{2}+4)}{3x^{2}+4x+5}}$ with respect to $x$.

If $y = x^{(x^x)}$,then $\frac{dy}{dx} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module